For the sifted bits, Alice publicly announces her choice of intensities for each WCP allowing Bob to tag each bit with its transmitted intensity. Let \(n_{X(Z),k}\)  denote the number of bits in the \(X(Z)\)  basis that originated from a pulse of intensity \(k\) . The secret key is constructed using only the sifted bits from the \(X\)  basis, while the bits from the \(Z\)  basis are used for parameter estimation. All the data from the \(Z\)  basis is publicly announced, which allows Alice and Bob to calculate the number of errors in this basis: \(m_{Z}\) . Additionally, they can calculate the number of errors in bits that originate from pulses with intensity \(k\) , \(m_{Z,k}\) . Using these results, one can calculate the phase error \(\phi _X\)  using Eq.s (4) and (5) of [1]}.
